In our life, very often we have to deal with the application of geometry in practice, for example, in construction. Among the most common geometric shapes there is also a trapezoid. And in order for the project to be successful and beautiful, a correct and accurate calculation of the elements for such a figure is necessary.
What is a trapezoid? This is a convex quadrangle that has a pair of parallel sides, called the base of the trapezoid. But there are two other sides connecting these foundations. They are called side. One of the questions regarding this figure is: "How to find the height of the trapezoid?" It is immediately necessary to pay attention that height is a segment that determines the distance from one base to another. There are several ways to determine this distance, depending on the known values.
1. The values of both bases are known, we denote them by b and k, as well as the area of this trapezoid. Using known values, it is very easy to find the height of the trapezoid in this case. As is known from geometry, the area of a trapezoid is calculated as the product of half the sum of the bases and height. From this formula, the desired value can easily be derived. To do this, you need to divide the area into half the amount of the grounds. In the form of formulas, it will look like this:
S = ((b + k) / 2) * h, hence h = S / ((b + k) / 2) = 2 * S / (b + k)
2. The length of the midline is known, we denote it by d, and the area. For those who do not know, the middle line is the distance between the midpoints of the sides. How to find the height of the trapezoid in this case? According to the trapezoid property, the middle line corresponds to half the sum of the bases, i.e. d = (b + k) / 2. Again, we resort to the area formula. Replacing half the sum of the bases with the value of the midline, we get the following:
S = d * h
As you can see from the resulting formula it is very easy to derive the height. Dividing the area by the value of the midline, we find the desired value. We write this with the formula:
h = S / d
3. The length of one side (b) and the angle formed between this side and the largest base are known. The answer to the question of how to find the height of the trapezoid is also in this case. Consider the trapezoid ABCD, where AB and CD are sides, with AB = b. The biggest reason is AD. The angle formed by AB and AD is denoted by α. From point B, lower the height h to the base AD. Now consider the resulting triangle ABF, which is rectangular. Side AB is hypotenuse, and BF-side. From the property of a right-angled triangle, the ratio of the leg value to the hypotenuse value corresponds to the sine of the angle opposite the leg (BF). Therefore, based on the foregoing, to calculate the height of the trapezoid, we multiply the value of the known side and the sine of the angle α. In the form of a formula, it looks like this:
h = b * sin (α)
4. The case is considered similarly if the size of the side and the angle are known, we denote it by β, which is formed between this side and the smaller base. In solving this problem, the angle between the known lateral side and the conducted height will be 90 ° - β. From the property of triangles - the ratio of the length of the leg and the hypotenuse corresponds to the cosine of the angle located between them. From this formula it is easy to derive the height value:
h = b * cos (β-90 °)
5. How to find the height of the trapezoid if only the radius of the inscribed circle is known? From the definition of a circle, it touches one point on each base. In addition, these points are in line with the center of the circle. It follows from this that the distance between them is the diameter and, at the same time, the height of the trapezoid. Looks like that:
h = 2 * r
6. Often there are problems in which it is necessary to find the height of an isosceles trapezoid. Recall that a trapezoid with equal sides is called isosceles. How to find the height of an isosceles trapezoid? With perpendicular diagonals, the height is equal to half the sum of the bases.
But what if the diagonals are not perpendicular? Consider the isosceles trapezoid ABCD. According to its properties, the bases are parallel. It follows that the angles at the bases will also be equal. Draw two heights BF and CM. Based on the foregoing, it can be argued that the triangles ABF and DCM are equal, that is, AF = DM = (AD - BC) / 2 = (bk) / 2. Now, based on the conditions of the problem, we determine the known values, and only then find height, considering all the properties of an isosceles trapezoid.